Finite Difference Model to Study the Effects of Na + Influx on Cytosolic Ca 2+ Diffusion

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1 World Academy of Science, Engineering and Technology Finite Difference Model to Study the Effects of + Influx on Cytosolic Ca + Diffusion Shivendra Tewari and K. R. Pardasani Abstract Cytosolic Ca + concentration in neurons is necessary for signal transduction. There are several parameters that affect its concentration profile like leaks, pumps, sources etc. Models and studies done so far have not included the effect of sodium influx on Ca + concentration which is a very important parameter needed to be taken into account in order to have a more realistic model. In this paper, we have incorporated the effect of + influx on [Ca + ] profile. Interdependence of all the important parameters like +, diffusion coefficient, time, and influx over [Ca + ] profile is studied. A program has been developed using finite difference technique for the entire problem and simulated on an AMD-Turion 64X machine to compute the numerical results. Keywords Ca + profile, + influx, Ca + chelators, Neurotransmitter release. I. INTRODUCTION ALCIUM [Ca + ] is an important second messenger, found Cin almost all cell types. [Ca + ] behaves like a switch in the process of signal transduction in which electrical signal is converted into a chemical signal. Several experiments have shown that a number of parameters affect the behavior of this second messenger. The parameters that can be held responsible are Voltage gated [Ca + ] channels, Ligand gated channels, Plasma Membrane Calcium ATPase (PMCA), + Ca + exchanger, Sarco-Endoplasmic Reticulum pump (SERCA), Mitochondrial stores, Endoplasmic Reticulum (ER) stores etc. Thus, in order to have a complete model for understanding the intracellular [Ca + ] behavior we should incorporate all the possible necessary parameters. The Mathematical models proposed so far have not incorporated the effect of + over cytosolic [Ca + ] profile (Neher, 986; Smith 996, 000; Rudiger et al. 007). Further, Reuter and Seitz (968) found that calcium extrusion in heart muscles is caused by the electrochemical sodium gradient across the plasma membrane. Blaustein (974) also observed that the sodium gradient across the plasma membrane influences the intracellular calcium concentration in a large variety of cells via a counter transport of + for Ca +. The dependence of + Ca + electrochemical gradient has been studied by Sheu and Fozzard (98) for sheep ventricular muscle and Purkine strands. Thus, there is enough evidence that + is an Shivendra Tewari is with the Maulana Azad tional Institute of Technology, Bhopal, MP 4605 INDIA (phone: ; fax: ; shivendratewari@manit.net.in). K. R. Pardasani, is with the Maulana Azad tional Institute of Technology, Bhopal, MP 4605 INDIA (phone: ; fax: ; kamalrap@rediffmail.com, kamalrap@hotmail.com). important parameter to be considered when modeling cytosolic [Ca + ] concentration. Fuioka et al. (000) also found that + Ca + is the maor mechanism by which cytoplasmic Ca + is extruded from cardiac myocytes. That is why, in our present mathematical model, we have incorporated + influx in order to study its effect over cytosolic [Ca + ] profile. + permeability, concentration and other parameters needed are taken as from Qian and Senowski (988). In this paper, we have used Goldman-Hodgkin-Katz current equation (Keener & Sneyd, 998) to frame for the + and Ca + flux. For the simulation of the proposed model, we have used finite difference method, Forward Time Centered Space (FTCS) technique. A computer program is developed in MATLAB and simulated on an AMD Turion 64X machine to obtain the numerical results. II. MATHEMATICAL MODEL In the present mathematical model, we have assumed that there is a voltage gated sodium channel adacent to the voltage gated calcium channel. It is assumed that [Ca + ] and [ + ] channels are always open, system reaches steady state within 00 ms (Smith, 996), length of the cytosol to be 4 μm and our system to be a homogenous system. Our mathematical model assumes the following reaction-diffusion kinetics (Neher, 986; Smith, 996), k [ Ca ] [ B ] [ CaB ] () where [B ] and [CaB ] are free and bound buffers respectively, and is an index over buffer species. The resulting partial differential equations for equation () using Fickian diffusion can be stated as (Smith, 996) where, k [ Ca ] DCa [ Ca ] R t () [ B ] DB [ B ] R t [ CaB ] D [ CaB ] R t CaB (3) (4) R k [ B ][ Ca ] k [ CaB ] (5) D Ca,, D B, D CaB are diffusion coefficients of free calcium, free buffer, and Ca + bound buffer, respectively; k + and k - are association and dissociation rate constants for buffer, 670

2 World Academy of Science, Engineering and Technology respectively. For stationary, immobile buffers or fixed buffers D B = D CaB = 0. Given, equation ( 5) can be further simplified, thanks to Neher (986), to the following: [ Ca ] DCa [ Ca ] k [ B] ([ Ca ] [ Ca ] ) t () r Along with the initial and boundary conditions as: t0 (6) A. Initial condition: [ Ca ] 0. M (7) B. Boundary conditions: dca [ ] lim DCar Ca dr r0 (8) lim[ Ca ] 0. M (9) r Here, [Ca + ] is the background [Ca + ] concentration, [B] is the total buffer concentration, Ca represents the flux due to [Ca + ]. As stated in equation (0) [Ca + ] tends to the background concentration of 0. μm as r but the domain taken by us is not infinite but a finite one. Thus, we have taken the distance required by [Ca + ] to attain background concentration to be 4 μm, as observed in experimental studies, (Allbritton et al., 99). Now, from GHK current equation (Keener & Sneyd, 998), we have, I P z FV m zfvm [ ] i [ ] oexp( ) zfvm ( exp( )) (0) where, I (C/s) is the current due to sodium gradient, P (m/s) is the sodium permeability, z is the valence of sodium ion (i.e. +), V m (volts) is the membrane potential, F (C/moles) is the Faradays constant, R (J/K mole) is the gas constant, T (K) is the absolute temperature, [ + ] i and [ + ] o are the intracellular and extracellular [ + ] (moles/l) concentration respectively. We have to convert equation (0) from C/s into moles/s using Faraday s constant. That is, the net flux of [ + ] ions flowing per second ( ) is, I () z F where, all the parameters have their usual meaning. In equation () there is a negative sign before I because inward current is taken to be negative. Similarly, using GHK current equation we can also get the expression for Ca i.e. the net flux of [Ca + ] from the voltage gated calcium channel. Before, substituting net flux of ions into equation () they were converted into μmolar / second by dividing net flux of ions by the volume of the cytosol and noting that L = 0 5 μm 3. Thus, incorporating these two expressions in eqn. (6), we have, [ Ca ] DCa [ Ca ] k [ B] ([ Ca ] [ Ca ] ) t Ca () So, our problem is to solve equation () coupled with equation (7 9). For our convenience we used the convention of writing u in lieu of [Ca + ]. We have applied finite difference scheme on equation () to get, ui u i ui ui u i ui u i DCa k h r i h (3) PCa ( uout ui e ) km[ B] ( ui u) ( e ) Here, FV m is a dimensionless quantity, h represents spatial step and k represents time step, i and represents the index of space and time respectively. Since, the above expression is not valid at the mouth of the channel; therefore the approximation at the mouth of the channel is given by (see appendix for details): 0 ( u pu p g kse sh sh (4) e p( q 0)) u0 p( q 0) uout f r DCa r DCa kdca h PCa p, q, [ ],, i g kkm B s where, h ri ( e ) f gu ksu k out Approximation for rest of the nodes is given by, ui p( qi) ui ( pgkse ) ui (5) p( qi ) ui f After doing stability analysis of our solution it was found that our solution is a mesh dependent solution which is stable for p 0.5. III. RESULTS AND DISCUSSION In this section, we have found the results corresponding to equation (4) and equation (5). The numerical values of parameters are as taken from Qian and Senowski (988). [Ca + ] permeability is calculated from Takeuchi (963), as per his observations [Ca + ] permeability is maximum when P Ca /P is in between 0.5 to 0.4. We have also assumed the same experimental environment as it was in the experiments done by Qian and Senowski (988). The parameters used are as stated in the Table I, otherwise specified with the figure. We have divided this section into five subsections: TABLE I VALUES OF BIOPHYSICAL PARAMETERS USED Symbol Parameter Value F Faraday s Constant Coulombs / Mole R Gas Constant 8.34 Joule / Kelvin Mole T Absolute Temperature 0 C P Ca Resting Permeability 9.05 x 0 - meter / second [Ca + ] P Resting Permeability 6.07 x 0-0 meter / second 67

3 World Academy of Science, Engineering and Technology [ + ] V m Resting membrane Volts potential [ + ] i Cytosolic [ + ] mm [ + ] o Extracellular [ + ] 45 mm [Ca + ] o Extracellular [Ca + ] mm D Ca Diffusion coefficient 50 μ meter / second + k m Buffer association rate.5 μ Molar - second - (EGTA) + k m Buffer association rate 600 μ Molar - second - (BAPTA) [B m] Buffer concentration 50 μ Molar F Faraday s Constant Coulombs / Mole R Gas Constant 8.34 Joule / Kelvin Mole A. WHEN BOTH [Ca+] AND [+] ARE FIXED B. WHEN [Ca + ] IS FIXED AND [ + ] IS VARIED Fig. The plot between Ca + and space for different instances of time. Parameters: As stated in the Table I In Fig., [Ca + ] is plotted against space for different time after channel opening. The distance is measured from the mouth of the channel. For t = ms, the results match with the results by Smith (996). But as time elapses it is getting harder for cytosolic [Ca + ] to maintain its background concentration and this is evident from the fall in the [Ca + ] curve for t = 00 ms after μm. Usually, it is believed that [Ca + ] attains its equilibrium around ~0- μm but we have taken this to be 4 μm because of the [ + ] influx. Fig. 3 Shows the plot between Ca + and time at μm for different instances of cytosolic +. Parameters: As stated in the Table I In Fig. 3, the effects of varying calcium were studied at a distance of μm from the [Ca + ] channel. It is assumed that our system is a homogenous system. The total amount of [ + ] in the system is 57 mm. There is no loss of [ + ] during its diffusion from outside of the cell to the inside. The studies were done for four cases: (i) When [ + ] o = 57 mm and [ + ] i = 0 mm (ii) When [ + ] o = 07 mm and [ + ] i = 50 mm (iii) When [ + ] o = 57 mm and [ + ] i = 00 mm (iv) When [ + ] o = 0 mm and [ + ] i = 57 mm The legends mentioned in Fig. 3 are for intracellular [ + ]. It is apparent from Fig. 3 that [Ca + ] concentration is reducing with respect to rising intracellular [ + ]. The changes became noticeable when we surpassed the benchmark of 0 ms. Similar studies are also done in Fig. 4. Fig. The plot between Ca+ and time at different distances from the source of the channel. Parameters: As stated in the Table I In Fig., results are plotted for [Ca + ] against time for different positions from the mouth of the channel (i.e. r = 0.5 μm, 0.5 μm, 0.75 μm,.0 μm). These results show no effect of the [ + ] influx, i.e. there is no effect of + influx on the timescale variation of Ca +. As expected these nodal concentrations of [Ca + ] are getting lower and lower as we are getting away from the channel. That is why, synaptic vesicles are close to the [Ca + ] channel (Tang et al., 000). Fig. 4 Shows the plot between Ca + and space for different instances of cytosolic + at time, t = 00 ms. Parameters: As stated in the Table I In Fig. 4, studies are done for the relation between varying [ + ] i and distance from the [Ca + ] channel. These studies were done inline with Fig. 3 so as to complete our spatiotemporal study of [Ca + ] with changing intracellular [ + ]. Again in this figure the effect of varying intracellular [ + ] are studied over cytosolic [Ca + ], only difference is that all the plots were made for time, t = 00 ms, instead of space. As shown in Fig. 4, for [ + ] i = 0 mm, [Ca + ] finds it very difficult to maintain its background concentration while for [ + ] = 57 mm the plots are as if there is no + influx. The 67

4 World Academy of Science, Engineering and Technology reason behind this is the chemical gradient of +, as we have supposed that when all the + is inside the cytosol, then there is no influx of + and also when there is no + inside the cytosol the + influx is maximum. impermeable to [Ca + ]. Whatever variations that are evident from the figure are because of the [ + ] influx. Fig. 5 shows the plot of Ca + against space for different cytosolic + concentration and different instances of time values. Parameters: As stated in the Table I Fig. 5 is also in sequel to Fig. 3 and Fig. 4, showing the effect of changing intracellular [ + ] with changing time (i.e. at t = ms, 0 ms and 00 ms). It is clear from Fig. 5(a) that cytosolic [Ca + ] is attaining its background concentration earlier than rest of the figures. This position of attaining background [Ca + ] is becoming more distant as intracellular [ + ] is decreasing to 0 mm, to be exact for [ + ] = 57 mm this place is at 6 μm while for [ + ] = 0 mm this place is at 4 μm (i.e. at boundary condition), both these values are for time, t = 00 ms. The cause for this is the fact that when all the [ + ] is present inside the cytosol the inflow of [ + ] is 0 and hence there is no effect over cytosolic [Ca + ]. Similarly, when intracellular [ + ] is absent, the inflow of [ + ] is maximum and hence the cytosolic [Ca + ] is not able to relax even till r = 4 μm. C. WHEN [ + ] IS FIXED AND [Ca + ] IS VARIED Fig. 7 The contour plot of Ca + with space and time. Parameters: As in Table I, except P Ca is equal to zero In Fig. 7, the result obtained in second part of Fig. 6 is shown using a contour plot. It is used to show the spatiotemporal variations in [Ca + ] in the absence of external [Ca + ] flux. It was observed that the highest [Ca + ] was 0.3 μm near the channel. It is known that synaptic vesicles are near the channels (Tang et al., 000) and at least three [Ca + ] must bind with a synaptic vesicles in order to initiate the process of exocytosis (Dodge & Rahamimoff, 967). Also, we know that, synaptotagmin is activated only at high cytosolic [Ca + ] ~ 0 μm and not at low concentrations [Ca + ] ~ μm (Brose et al., 99). Thus, we can conclude that at the given uncture it is impossible to have any significant amount of neurotransmitter release for signal transduction. D. [Ca + ] AGAINST [ + ] Fig. 6 The plot between Ca + and space at different instance of time t. Parameters: As stated in the Table I except for the left figure [Ca + ] o is absent and for the right P Ca is set to zero Fig. 6, shows the relationship between external [Ca + ] and [Ca + ] flux over cytosolic [Ca + ]. Therefore, two simulations were made while keeping all the parameters as stated in Table except for the first simulation setting external [Ca + ] to zero and for the second simulation setting [Ca + ] permeability to zero. These simulations gave identical results, because when external [Ca + ] is absent there is no [Ca + ] gradient across the cytosol membrane and hence no [Ca + ] flows inside the cytosol. In the second figure, it was done manually by setting [Ca + ] permeability to zero i.e. making the cytosol membrane Fig. 8 [Ca + ] is plotted against [ + ]. Before plotting the results the data was normalized, so that it can be compared for different distances from source. Parameters: As stated in the Table I. In Fig. 8, we have plotted varying cytosolic [Ca + ] against varying cytosolic [ + ]. The values are plotted for different values of r i.e. distance from source. We have normalized the [Ca + ] concentration obtained for different values of r. It is evident from the figure that the slope of [Ca + ] for [ + ] is almost negligible because when we are close to the channel there is not much of a change happening with respect to increasing cytosolic [ + ]. Also, as we move away from the source i.e. between 3.3 μm to 6.6 μm, the slope gets steeper and steeper, because the impact of [Ca + ] channel is getting weak and impact of [ + ] is getting clear. But it can be observed from the figure that the increase in slope for r = 6.6 μm and r = 8.3 μm is not much, it is because as we move 673

5 World Academy of Science, Engineering and Technology further from the channel the impact of [ + ] is also getting weak. E. [Ca + ] AGAINST VARYING BUFFER CAPACITY The results plotted in this section are quite common and are in agreement with the results obtained by previous researchers (Neher, 986; Smith, 996; Smith et al., 000). It ust contains two figures one for increasing buffer concentration and another for different buffer capacity. Fig. 9 Ca + is plotted against space for different buffer concentrations at time t = 00ms. Parameters: same as in Table I except the buffer concentration. In Fig. 9, graph is plotted for EGTA. We have shown the effect of increasing buffer concentration over cytosolic [Ca + ]. These buffers are exogenous in nature and are used to slow the diffusion process. Apparent from the figure, as buffer concentration is increased the free [Ca + ] is decreasing with space and thus background concentration is being achieved at an early stage. useful for biomedical scientists for development of new protocols for treatment and diagnosis of neurological diseases. APPENDIX Using Laplacian operator in spherical symmetry, we have r r r (6) Further, we have used Forward Time Centered Space (FTCS) technique to solve eqn. () i.e. u u t du u dr i ui k i ui (7) h d u ui ui ui dr h Using equation (7) in equation () and solving we get equation (5) i.e. u p( q ) u ( pgkse ) u i i i i p( qi ) ui f But this approximation does not hold for i = 0, as it gives rise to an imaginary node. To eliminate this problem, we have used centered difference over equation (8) to yield, sh u u ( u 0e uout ) r0 DCa Here, the scale used for distance was [0.000, 4.000] μm. Thus for i = 0, we have equation (4) i.e. u 0 pu (pgkse sh sh e p( q 0)) u0 p( q 0) uout f r D r D Ca Ca Since, the net influx of [Ca + ] and [ + ] ions was in moles / second therefore, it was converted into μmoles / second before being used in equation (4) and (5). ACKNOWLEDGMENT Fig. 0 Ca + is plotted against space for different buffers. Parameters: same as in Table I except for the buffers In Fig. 0, we have shown the effect of two different exogenous buffers EGTA and BAPTA. The fast chelator BAPTA s swiftness is apparent from the figure. There is not much that is needed to be said about these buffers as already a lot is known about these buffers. Some of the results that are shown in this paper are in agreement with results obtained by previous researchers (Neher, 986; Smith, 996; Smith et al., 000). Whatever new results that are posted are also in agreement with the physiological facts. The results obtained in this paper may be The authors are highly grateful to Department of Biotechnology, New Delhi, India for providing support in the form of Bioinformatics Infrastructure Facility for carrying out this work. REFERENCES [] N. L. Allbritton, T. Meyer, and L. Stryer, Range of messenger action of calcium ion and inositol,4,5-trisphosphate, Science, vol. 58, pp. 8 85, 99. [] M. P. Blaustein, The interrelationship between sodium and calcium fluxes across cell Membranes, Rev. Physiol. Biochem. Pharmacol., vol. 70, pp. 33-8, 974. [3] M. P. Blaustein and A. L. Hodgkin, The effect of cyanide on the efflux of calcium from squid axons, J. Physiol., vol. 00, pp ,

6 World Academy of Science, Engineering and Technology [4] N. Brose, A. G. Petrenko, T. C. Sudhof, and R. Jahn, Synaptotagmin: a calcium sensor on the synaptic vesicle surface, Science, vol. 56, pp. 0-05, 99. [5] F. A. Dodge r and R. Rahamimoff, Co-operative action of calcium ions in transmitter release at the neuromuscular unction, J. Physiol., vol. 93, pp , 967. [6] E. M. Fenwick, A. Marty, and E. Neher, Sodium and calcium channels in bovine chromaffin cells, J. Physiol., vol. 33, pp , 98. [7] Y. Fuioka, K. Hiroe, and S. Matsuoka, Regulation kinetics of +- Ca+ exchange current in guinea-pig ventricular myocytes, J. Physiol., vol. 59, pp. 6-63, 000. [8] J. Keener and J. Sneyd, Mathematical Physiology, Vol. 8, Springer, 998, pp [9] E. Neher, Concentration profiles of intracellular Ca+ in the presence of diffusible chelators, Exp. Brain Res. Ser., vol. 4, pp , 986. [0] N. Qian and T. J. Senowski, Electrodiffusion Model of Electrical Conduction in Neuronal Processes, Cellular Mechanism of Conditioning and behavioral plasticity, Plenum publishing corporation, pp , 988. [] H. Reuter and N. Seitz, The dependence of calcium efflux from cardiac muscle on temperature and external ion composition, J. Physiol., vol. 95, pp , 968. [] S. Rüdiger, J. W. Shuai, W. Huisinga, C. gaiah, G. Warnecke, I. Parker, and M. Falcke, Hybrid Stochastic and Deterministic Simulations of Calcium Blips, Biophysical J., vol. 93, pp , 007. [3] S. S. Sheu and H. A. Fozzard, Transmembrane + and Ca+ Electrochemical Gradients in Cardiac Muscle and Their Relationship to Force Development, J. Physiol., vol. 80, pp , 98. [4] G.D. Smith, Analytical Steady-State Solution to the rapid buffering approximation near an open Ca+ channel, Biophys. J., vo. 7, pp , 996. [5] G.D. Smith, L. Dai, R. M. Miura and A. Sherman, Asymptotic Analysis of buffered Ca+ diffusion near a point source, SIAM J. of Applied of Math, vol. 6, pp , 000. [6] N. Takeuchi, Effects of calcium on the conductance change of the endplate membrane during the action of transmitter, J. Physiol., vol. 67, pp. 4-55, 963. [7] Y. Tang, T. Schlumpberger, T. Kim, M. Lueker, and R. S. Zucker, Effects of Mobile Buffers on Facilitation: Experimental and Computational Studies, Biophys. J., vol. 78, pp ,